Work done by a spring We know that work equals force times displacement But how to we calculate the work due to a nonconstant force Reconsider the restoring force of a spring Fs kx Hooke s Law for the restoring force of an ideal spring It depends on the distance the spring is stretched or compressed d x f xi But the force is not constant Fs i kxi Fs f kx f xi Take the average force xf Fs avg Fs avg x 0 Fs i Fs f 2 1 2 k x f x i Then the work done by the spring is W s Fs avg cos d Fs avg cos0 d 1 2 1 2 2 f 2 i k x f x i x f x i k x x 1 2 2 i 1 2 W s kx kx 2 f 2 i 2 f Ws kx kx U s i U s f 1 2 1 2 U elastic U s kx 1 2 2 Total potential energy is Units of N m m2 Nm J U total U g U s mgy kx 1 2 2 Example Problem A block m 1 7 kg and a spring k 310 N m are on a frictionless incline 30 The spring is compressed by xi 0 31 m relative to its unstretched position at x 0 and then released What is the speed of the block when the spring is still compressed by xf 0 14 m x 0 xi x 0 xf Given m 1 7 kg k 310 N m 30 xi 0 31 m xf 0 14 m frictionless Method no friction so we can use conservation of energy Initially 2 E mv mgh kx v i 0 hi xi sin 1 2 1 2 2 i Ei mgxi sin kx 1 2 2 Finally h f x f sin find v f 2 2 1 1 E f 2 mv f mgx f sin 2 kx f E f Ei 2 2 1 1 2 mv f mgx f sin 2 kx f 2 1 mgxi sin 2 kxi 2 2 2 1 1 2 mv f mg xi x f sin 2 k xi x f k 2 2 v f 2 g xi x f sin xi x f m 310 v f 2 9 8 0 31 0 14 sin 30 0 312 0 14 2 1 7 v f 1 666 13 95 3 95 ms Interesting to plot the potential energies Energy K Ug xf E Utotal Us xi xf xi xf xi
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